# Finding the general solution to a differential equation

• clarineterr
In summary, the conversation discusses finding a particular solution to the equation \frac{d^{2}y}{dt} +4\frac{dy}{dt}+20y=e^{-2t}(sin4t+cos4t). The solution to the homogeneous equation is given and a possible solution for the non-homogeneous term is suggested, which is later confirmed to work.
clarineterr

## Homework Statement

$$\frac{d^{2}y}{dt}$$ +4$$\frac{dy}{dt}$$+20y=e$$^{-2t}$$(sin4t+cos4t)

## The Attempt at a Solution

The solution to the homogeneous equation: $$\frac{d^{2}y}{dt}$$ +4$$\frac{dy}{dt}$$+20y=0 is

y= k1e$$^{-2t}$$cos4t +k2e$$^{-2t}$$sin4t

Then I guessed ae$$^{-2+4i}$$ as a possible solution and it didn't work, and that's where I'm stuck.

Since the complementary solution yc= e-2t(Acos(4t) + Bsin(4t)) is repeated in the non-homogeneous term, for your particular solution try

yp = te-2t(Ccos(4t) + Dsin(4t))

Nope...didn't work. :(

clarineterr said:
Nope...didn't work. :(

Yes, it does work. Check your work or show it here if you can't find your error.

## 1. What is a differential equation?

A differential equation is a mathematical equation that describes the relationship between a function and its derivatives. It involves the use of variables, functions, and their derivatives to model various physical phenomena, such as the growth of populations, the motion of objects, and the flow of fluids.

## 2. What is the general solution to a differential equation?

The general solution to a differential equation is a family of solutions that satisfies the equation for all possible values of the independent variables. It contains an arbitrary constant that can take on any value, allowing for a wide range of possible solutions.

## 3. How do you find the general solution to a differential equation?

To find the general solution to a differential equation, you need to integrate both sides of the equation with respect to the independent variable. This will eliminate the derivatives and leave you with an equation that contains only the original function and an arbitrary constant. The general solution can also be found by using methods such as separation of variables, substitution, and integration by parts.

## 4. What is the difference between a general solution and a particular solution?

A general solution is a family of solutions that contains an arbitrary constant, while a particular solution is a specific solution that satisfies the differential equation for given initial conditions. The general solution represents all possible solutions, while a particular solution represents a specific solution within that family.

## 5. Can a differential equation have multiple general solutions?

No, a differential equation can only have one general solution. However, this general solution may contain multiple arbitrary constants, allowing for a wide range of possible solutions. Multiple particular solutions can be derived from the general solution by assigning specific values to the arbitrary constants.

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